Is there already literature where this is treated? I tried Ladyschenkaja but there this type of system is not present (I believe in that book, they require $a_1 \equiv a_5$). I would appreciate if anyone had any references to this problem.

I believe though that I can probably apply a fixed-point argument to this -- again I would appreciate if someone gave a pointer as where such things are discussed. Thanks.

2 Answers
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There is a classical counterexample due to Plis of an elliptic differential operator with Hölder continuous coefficients without Cauchy uniqueness. This was refined with a counterexample in divergence form by Miller in a 1974 Arch. Rat. Mech.(vol. 54) article for the elliptic and parabolic case.

Hölder continuity is not enough to get uniqueness results for parabolic or elliptic equations.

A good reference is Krylov's Lectures on elliptic and parabolic equations in Hölder spaces. On another note, if $a_1\not\equiv a_5$ is the only reason you are stuck, then it should be possible to adapt the arguments you have for $a_1\equiv a_5$ to $a_1\not\equiv a_5$, by using uniform ellipticity.